The bilinear form \(a(\cdot ,\cdot )\)  satisfies the coercivity condition \(a(u,u)\ge C_{\Omega }\Vert u\Vert _{H_0^1(\Omega )}^2\)  for any \(u\in H_0^1(\Omega )\)  and \(q\in K\) , where \(C_{\Omega }\)  is a positive constant depending on \(\Omega \)  and the low bound \(a_0\)  of \(q\) . By the Lax-Milgram theorem [1]}, there exists a unique weak solution \(u\)  of (REF ) in \(H_0^1(\Omega )\)  for each \(q\in K\) , and \(u\)  is nonlinearly dependent on \(q\) . Then, we can define the nonlinear coefficient-to-solution mapping \(U:K\rightarrow H_0^1(\Omega )\) , which maps each \(q\in K\)  to the unique solution \(u=U(q)\in H_0^1(\Omega )\)  of (REF ); see [2]} for more details.
